Problem 1: (20 points)
Construct a PDA for each of the following languages:
a) L={a ³ⁿ b ⁿ | n > 0}. Construct the PDA without going through grammars.
b) L={a ⁿ b ^n+m a ^m | n > 0, m> 1}. Again, do not go through grammars.
c) The languages in Problem 2 in Homework 3. For these languages, use the GNF grammars (developed in Problem 5 of Homework 3) to construct the PDA's using the standard method.

Problem 2: (24 points)
Show that the following languages are not CFL's.
a) L={a ^{n 3} | n > 0}
b) L={a ^p | p is a prime number}.
c) L={a ⁿ b ^m c ^m | m>n}.
d) L={a ^n! | n> 0}.
e) L={a ⁿ b ^{n 2} | n > 0}.
f) L={a ⁿ b ^m c ^{n 2 +m 2} | n > 0, m > 0}.

Problem 3: (16 points)
Using the closure properties of CFL's, show that the following languages are CFL's:
a) L={a ⁿ b ⁿ | n > 0 and n is not a multiple of 3}
b) L={a ⁱ b ^j | i < j or i=j+2}
c) L={a ⁱ b ^j c ^k | i < j or j < k}
d) L= the complement of {a ⁿ b ⁿ | n > 0}.

Problem 4: (20 points)
Design Turing machines to recognize the following languages. Your design (in this Problem and in Problem 5) should start with a clear algorithmic description and end with a formal definition of the delta function (with enough comments).
a) {a ²ⁿ b ⁿ | n > 0}.
b) {a ⁿ b ^m c ^n+m | n>0, m>0}.

Problem 5: (20 points)
a) Build a TM that takes as input 0ⁿ and gives output 0 ²ⁿ.
b) Build a TM that takes as input 0ⁿ10^m and gives output 0 ^max(n,m) .